Finite Group Databasebeta

← C3⋊#23778468 #44478052

Subgroups — C3⋊#23778468 #44478052

Order 30420.

148 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5⋊#11759817C211yesmaximal, characteristic
2C13×C3⋊#63121C311yesmaximal, characteristic, complemented
3#23778468—31nomaximal
4C3×C5×#92788C411yescharacteristic, complemented, Hall
5C3⋊#14523152—51nomaximal, Hall
6C3×C13×D65C2×C311yescharacteristic
7C5⋊#11759817—21no—
8C5⋊#11759817D311yescharacteristic
9C13×#63121—31noHall
10C3×#11759817—51no—
11C5×#92788—21no—
12C5×#92788Dic311yescharacteristic, complemented
13C3×C5×C132C3×C411yescharacteristic, complemented
14C13⋊C3×ΓL(1,𝔽16)—131nomaximal
15C3⋊(#63121⋊C3)—131nomaximal
16C13×C3⋊#261—51no—
17#14523152—151no—
18C13×D65C3×D311yescharacteristic
19C3×#92788AGL(1,𝔽5)11yescharacteristic, complemented, Hall
20C3×C5⋊(D13⋊C3)—131no—
21C3×D5×C13⋊C3—131no—
22C3×C13×D13—51no—
23#11759817—51no—
24#11759817—101no—
25C5×C132C3×Dic311yescharacteristic, complemented, Hall
26C3⋊#63121C13⋊C311yescharacteristic, complemented
27C13×ΓL(1,𝔽16)—131no—
28#63121⋊C3—391no—
29AGL(1,𝔽5)×C13⋊C3—391no—
30C13×#261—151noHall
31C3×C5×C13⋊C3—131no—
32C3×C5×C13⋊C3—523no—
33C3×C5×C13⋊C3—521no—
34C3×C132C3×AGL(1,𝔽5)11yescharacteristic, complemented
35#92788—21no—
36#92788ΓL(1,𝔽16)11yescharacteristic, complemented
37C3⋊AGL(1,𝔽13)—651no—
38Dic3×C13⋊C3—651no—
39C3×D65C2×C13⋊C311yescharacteristic
40C3×D5×C13—131no—
41C5⋊(D13⋊C3)—131no—
42C5⋊(D13⋊C3)—261no—
43D5×C13⋊C3—131no—
44D5×C13⋊C3—261no—
45C13×D13—51no—
46#63121—31no—
47C13×AGL(1,𝔽5)—391no—
48C2×C3×C13⋊C3—651no—
49C3×D13⋊C3—651no—
50C3×C5×C13—43no—
51C3×C5×C13C4×C13⋊C311yescharacteristic, complemented
52C3×C5×C13AGL(1,𝔽13)11yescharacteristic, complemented
53C5×C13⋊C3—131no—
54C5×C13⋊C3—261no—
55C5×C13⋊C3—526no—
56C5×C13⋊C3—523no—
57C5×C13⋊C3—521no—
58C5×C13⋊C3—521no—
59C3×ΓL(1,𝔽16)—1691noHall
60C132C3×ΓL(1,𝔽16)11yescharacteristic, complemented, Sylow
61C3⋊#261—51no—
62AGL(1,𝔽13)—1951no—
63Dic3×C13—651no—
64C4×C13⋊C3—1951no—
65D65D3×C13⋊C311yescharacteristic
66D5×C13—131no—
67C3×C13⋊C3—131no—
68C3×C13⋊C3—523no—
69C3×C13⋊C3—521no—
70C32×D5—1691no—
71C2×C3×C13—651no—
72C3×D13—51no—
73C2×C13⋊C3—651no—
74C2×C13⋊C3—1301no—
75D13⋊C3—651no—
76D13⋊C3—1301no—
77C5×C13—43no—
78C5×C13Dic3×C13⋊C311yescharacteristic, complemented
79C5×C13C3⋊AGL(1,𝔽13)11yescharacteristic, complemented
80ΓL(1,𝔽16)—131no—
81C3×AGL(1,𝔽5)—5071no—
82#261—151no—
83C4×C13—1951no—
84C32×C5—1691noHall
85C3×C13—43no—
86C3×C13AGL(1,𝔽5)×C13⋊C311yescharacteristic, complemented
87C3×C13#63121⋊C311yescharacteristic, complemented
88C13⋊C3—131no—
89C13⋊C3—261no—
90C13⋊C3—526no—
91C13⋊C3—523no—
92C13⋊C3—521no—
93C13⋊C3—521no—
94C3×Dic3—8451noHall
95C3×D5—131no—
96C3×D5—3381no—
97C3×D5—1691no—
98C2×C13—651no—
99D13—51no—
100AGL(1,𝔽5)—391noHall
101C2×C32—8451no—
102C3×C5—6761no—
103C3×C5—6761no—
104C3×C5#1452315211yescharacteristic, complemented
105C13—43no—
106C13C3⋊(#63121⋊C3)11yescharacteristic, complemented
107C13C13⋊C3×ΓL(1,𝔽16)11yescharacteristic, complemented
108C3×C4—25351no—
109Dic3—651no—
110D5—131no—
111C32—6761noSylow
112C2×C3—651no—
113C2×C3—8451no—
114C2×C3—16901no—
115C5C3⋊#1452315211yescharacteristic, complemented, Sylow
116C4—1951noSylow
117C3—6761no—
118C3—6761no—
119C3#2377846811yescharacteristic, complemented
120C2—651no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C132C3×C4
prime radicalC3×C5⋊#11759817C2
cosocleC3×C13×D65C2×C3
LCSG ▷ C3×C5×C132
UCSC1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C13×C3⋊#63121C3semidirect
4. C3×C5×#92788C4semidirect
12. C5×#92788Dic3semidirect
13. C3×C5×C132C3×C4semidirect
19. C3×#92788AGL(1,𝔽5)semidirect
25. C5×C132C3×Dic3semidirect
26. C3⋊#63121C13⋊C3semidirect
34. C3×C132C3×AGL(1,𝔽5)semidirect
36. #92788ΓL(1,𝔽16)semidirect
51. C3×C5×C13C4×C13⋊C3semidirect
52. C3×C5×C13AGL(1,𝔽13)semidirect
60. C132C3×ΓL(1,𝔽16)semidirect
78. C5×C13Dic3×C13⋊C3semidirect
79. C5×C13C3⋊AGL(1,𝔽13)semidirect
86. C3×C13AGL(1,𝔽5)×C13⋊C3semidirect
87. C3×C13#63121⋊C3semidirect
104. C3×C5#14523152semidirect
106. C13C3⋊(#63121⋊C3)semidirect
107. C13C13⋊C3×ΓL(1,𝔽16)semidirect
115. C5C3⋊#14523152semidirect
119. C3#23778468semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C5⋊#117598172
C13×C3⋊#631213
#237784683
C3⋊#145231525
C13⋊C3×ΓL(1,𝔽16)13
C3⋊(#63121⋊C3)13